Question
A rental kiosk has one portable power pack left before its evening pickup window. A dealer offers $200 now to take it, but the offer expires immediately. The kiosk also has a standing obligation to a contract customer: with probability 30% the customer will arrive tonight and pay $120 for the pack. If the kiosk cannot supply it, it must instead pay the customer $240 in contractual cancellation compensation. With probability 70% the customer does not arrive. If the kiosk keeps the pack and the customer never comes, it can sell it to a recycler after the pickup window for $50.
There is no other inventory, reacquisition option or cost, and all stated amounts are net cash from this decision. Management imposes a hard risk rule: terminal cash from the decision must never be below −$20, whichever customer-arrival state occurs.
Should the kiosk accept the dealer's offer or retain the pack? First compare expected cash outcomes, then enforce the risk rule rather than relying on expected value alone.
Get hint
Hint 1
Make a two-state cash table for each action: the contract customer arrives or does not arrive.
Check your answer
Solution
There are two possible states: the contracted customer arrives with probability 0.30, or does not arrive with probability 0.70. Selling now to the dealer pays $200 immediately, but leaves no unit to fulfil the commitment. In the arrival state the kiosk must pay $240 compensation, so its final cash is \(200-240=-\$40\). If no customer arrives, it keeps the $200. Expected cash from selling is
\[
E[C_{\rm sell}]=0.30(-40)+0.70(200)=\boxed{\$128}.
\]
Keeping the unit yields $120 in the arrival state, because the kiosk completes the promised rental, and $50 in the no-arrival state, because it disposes of the unused pack. Its expected cash is
\[
E[C_{\rm keep}]=0.30(120)+0.70(50)=\boxed{\$71}.
\]
Without the risk rule, the seller would accept the dealer's offer: $128 is $57 higher than $71. But the constraint requires \(C\ge-20\) in every state. Selling fails in the customer-arrival state (\(-40<-20\)); keeping produces at least $50, and so passes both states. The only feasible choice is therefore \(\boxed{\text{keep the pack}}\).
The conclusion is not based on average customer demand or on the average cancellation cost. A hard terminal shortfall limit vetoes an action as soon as one permitted state breaches it, even if that state is relatively unlikely and the action has higher expected cash.
Assumptions
The customer's 30% arrival probability is fixed and represents the only uncertainty.
The $240 contractual compensation is owed only if the customer arrives but the pack cannot be supplied.
Any unsold retained pack can be disposed of for $50 after the customer-arrival opportunity closes.
Net terminal cash is measured relative to zero immediately before the decision; minimum allowed cash is −$20.
Final Answer
Retain the pack. The dealer transfer has higher expected cash ($128 versus $71), but its −$40 outcome if the contract customer arrives violates the −$20 worst-case floor.